91Ó°ÊÓ

An autonomous system is a set of equations where the derivatives of the variables are functions of the variables themselves, not explicitly involving another independent variable like time. In our original problem with the differential equation \( x^{\prime \prime}+x-\epsilon x|x|=0 \), we convert it to a first-order system by introducing the variable \( y \) such that \( x' = y \). This approach results in:

• \( x' = y \)
• \( y' = -x + \epsilon x|x| \)

This system does not explicitly involve time, hence it is labeled a plane autonomous system. Here, 'plane' denotes the two-dimensional space created by the variables \( x \) and \( y \). Autonomous systems are useful for analyzing stability and behavior of systems through phase portraits, without considering external time factors.

Critical points in a system occur where the rate of change of all variables is zero. For our system:

• \( x' = y = 0 \)
• \( y' = -x + \epsilon x|x| = 0 \)

Setting \( x' = y = 0 \) simplifies finding critical points. With \( y = 0 \), substitution into \( y' = -x + \epsilon x |x| = 0 \) leads us to solve \( -x + \epsilon x^2 = 0 \). Factoring and solving gives the critical points:

• \( (0, 0) \)
• \( \left(\frac{1}{\epsilon}, 0\right) \)

These points indicate where the system is in equilibrium, meaning from these points, the system would not change state if undisturbed. Analyzing critical points helps in understanding the long-term behavior and stability of a dynamical system.

A second-order differential equation involves the second derivative of a function. In the exercise, the equation \( x^{\prime \prime} + x - \epsilon x|x| = 0 \) is of second order because \( x'' \) appears in it. Typically, such equations represent systems where acceleration or curvature is involved, like harmonic oscillators. To solve or analyze them, it is often helpful to reduce the order of the equation by expressing it as a system of first-order equations, which simplifies the complexity and allows for more straightforward analysis. Converting into first-order systems is a powerful method because it often reveals the underlying structure and symmetry of the system through matrices or geometric considerations.

A first-order system of equations is comprised of equations with only first derivatives. In the solution provided, we transformed the second-order equation into this type by using a substitution:

• Introduce \( y = x' \)
• Write \( x'' \) as \( y' \)

The resulting system:

• \( x' = y \)
• \( y' = -x + \epsilon x|x| \)

Such systems are advantageous because they allow for the application of linear algebra techniques and visualization through phase plane analysis. The dimensionality reduction from higher-order derivatives to first-order makes assessing behavior over time more approachable and is particularly useful for finding critical points and evaluating system stability. This transformation underlies many modeling techniques in physics and engineering.